"Pentagon" and "pentagram"

The word of "pentagon" (from Greek "pentagonon") is well known to us from the name of the U.S. Military Department building, which has in the plane the form of the regular pentagon (Fig. 1).

'Pentagon' and 'pentagram'
Figure 1. "Pentagon" and "pentagram".

However the figure on Fig.1 has also other name of "pentagram" (from the Greek words "pentagrammon", "pente " - 5 and "gramma " - a line) that means an regular pentagon, on which sides the isosceles triangles of identical altitude are constructed.

The diagonals of the "pentagon" derivate the "pentagonal star". It was proved that the intersection points of the diagonals always are the points of their "golden section". They derivate the new "pentagon" FGHKL. In the new "pentagon" it is possible to draw the diagonals, which intersection points derivate one more "pentagon" and this process can be continued ad infinitum. Thus, the "pentagon" ABCDE as though consists of infinite number of "pentagons", which are derivated by the intersection points of diagonals. This infinite repeatability of the same geometrical figure creates feeling of a rhythm and harmony, which is fixed by our reason unconsciously.

In the "pentagram" it is possible to find a huge number of the golden proportions. For example, the ratio of pentagon's diagonal to its side is equal to the golden proportion.

Let's consider now the sequence of the line segments FG, EF, EG, EB. It is easy to show, that they are connected by the following ratio:

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The "pentagram" always evokes Pythagorean delight and was considered by them as their main recognition symbol. There is the following legend. When the foreign land one of the Pythagoreans lied on a bier and could not pay to the person who served him he asked him to figured the "pentagram" on the dwelling hoping that this symbol will be seen by someone of the Pythagoreans. And really, some years after one Pythagorean saw this symbol and the house host got a rich reward.

The pentagram in Fig.1 includes a number of remarkable figures, which were used widely in works of art. In the ancient art the law of the "golden cup" (Fig.2) is widely known. This one used by antique sculptors and by the "golden" business craftsmen. The shaded part of the "pentagram" in Fig.2 gives a presentation of the "golden cup".

'Golden cup''Golden' triangle
Figure 2. "Golden cup".Figure 3. "Golden" triangle.

The "pentagonal star" included into "pentagram", consists of five equilateral "golden" triangles; each of them reminds a letter of "A" ("five intersected A") (Fig.3).

Each "golden" triangle has an acute angle A = 36° at the top and two acute angles D = C = 72° at the basis of the "golden" triangle. The main feature of the "golden" triangle consists of the fact that the ratio of the edge AC = AD to the basis DC is equal to the "golden" proportion t. Studying the "pentagram" and the "golden" triangle the Pythagoreans were admired when they have found out, that the bisectrix DH coincides with the diagonal DB of the "pentagon" (Fig.1) and divides the side AC in the point H by the golden section (Fig.3). Here there is the new "golden" triangle DHC. If now to draw the bisectrix of the angle H to the point H' and to continue this process ad infinitum, we will get an infinite sequence of "golden" triangles. As well as in the case of the "golden" rectangle and "pentagram" the infinite repetition of the same geometrical figure ("golden" triangle) after realization of the next bisectrix produces an aesthetic feeling of beauty and harmony.